Quantum Information Theory with Gaussian Systems

= − 1
2
2Nf
i,j=1
where M ′ =
M ′ i,j (α′ − β ′ )i (α ′ − β ′ )j ,
Nf
n=1
2 log
γn − 1
.
γn + 1
5.2 Security estimation
The last identity is due to the fact that T(α), T(β) and T(0) all possess the same
covariance matrix γ = 2Nf + σ T G−1 σ by (5.2) and that T(0) is centered around
zero, i.e. tr[T(0)R ′ k ] = 0 for all field operators R′ k with k = 1, . . .,2Nf. Recall from
Section 2.2.2 that the prime indicates the basis in which the covariance matrix is
diagonal. For isotropic, uncorrelated Gaussian noise with G = 2Nf/g this yields
γ = (1 + g) 2Nf with symplectic eigenvalues γn = 1 + g and thus
S T(α) T(β) = log(1 + 2/g) |α − β| 2 /2 ≤ 4 log(1 + 2/g)NfEmax .
Combining this estimate with (5.6) yields the bound
T(α) − T(β) 1 ≤ 2 2 log(1 + 2/g)NfEmax , (5.7)
which proves the functioning of the continuous randomization in the first place,
since both output states can be made arbitrarily indistinguishable from each other
by choosing g large enough.
As a first step towards the discrete protocol, we approximate the ideal randomization
(5.1) by the cutoff integral (5.3). To estimate the error T [](α) − T(α) 1 we
compare both channels with the nonnormalized, completely positive map c [ ]
c T [](α):
T[](α) − T(α) 1 ≤ T[](α) − c [ ]
c T [](α) 1 + c []
c T [ ](α) − T(α) 1 . (5.8)
Both terms will be estimated by the same bound for the difference between the full
and the cutoff classical integral:
T[ ](α) − c [ ]
c T [ ](α) = 1 1
c |c − c [ ]| T[](α)
1
= 1
dξ e
c
−ξT
·G·ξ/4
− dξ e −ξT
·G·ξ/4
c []
c T [](α) − T(α) 1 =
since T[](α) = 1 1
= 1
c
1
c
= 1
c
|ξ|≥a
|ξ|≥a
|ξ|≥a
|ξ|≤a
dξ e −ξT ·G·ξ/4 , (5.9a)
dξ e −ξT ·G·ξ/4
Wξ |α〉〈α|W ∗ ξ
1
dξ e −ξT ·G·ξ/4
since Wξ |α〉〈α|W ∗
ξ = 1.
1
(5.9b)
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